#848 ⟨a, b, c | ab=c, aacc=1⟩

Properties

Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a²(ba)² = a and a ⋅ 1 = a, however a²(ba)² ≠ 1
Not right cancellative, because right multiplication by a(ab)² is not injective:
- a²(ba)² ⋅ a(ab)² = a(ab)² and 1 ⋅ a(ab)² = a(ab)², however a²(ba)² ≠ 1

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(a) = deg(b) = 0, a < b; deg(c) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a³bab ⇒ 1	[2]
2.	c ⇒ ab	[1]

# abc:ab=c,aacc=1 ab/c - -
aaabab=1
c=ab

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

12 total

Σ	#	Presentation	Mapping
8	2338	⟨a, b, c \| aaa=b, bcac=1⟩	φ(a) = a, φ(b) = aaa, φ(c) = b
8	2497	⟨a, b, c \| aab=c, acab=1⟩	φ(a) = a, φ(b) = b, φ(c) = aab
8	2636	⟨a, b, c \| aba=c, aacb=1⟩	φ(a) = a, φ(b) = b, φ(c) = aba
8	2661	⟨a, b, c \| aba=c, bbbc=1⟩	φ(a) = b, φ(b) = a, φ(c) = bab
8	4200	⟨a, b, c \| aab=1, acac=b⟩	φ(a) = a, φ(b) = abab, φ(c) = b
8	4500	⟨a, b, c \| abc=1, aaca=b⟩	φ(a) = a, φ(b) = aaba, φ(c) = b
8	4687	⟨a, b, c \| aa=b, abcac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	4724	⟨a, b, c \| aa=b, bacac=1⟩	φ(a) = a, φ(b) = aa, φ(c) = b
8	4988	⟨a, b, c \| ab=c, aaabc=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5001	⟨a, b, c \| ab=c, aacab=1⟩	φ(a) = a, φ(b) = b, φ(c) = ab
8	5076	⟨a, b, c \| ab=c, bbbca=1⟩	φ(a) = b, φ(b) = a, φ(c) = ba
8	6824	⟨a, b, c \| ab=1, aacac=b⟩	φ(a) = a, φ(b) = aabab, φ(c) = b